1. 1 Designing Knowledge Scaffolds to Support Mathematical Problem Solving Rittle-Johnson, B., Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problem solving. Cognition and Instruction, 23(3), 313-349. 指導教授： Chen, Ming-puu 報 告 者： Jheng, Cian-you 報告日期： 2007/03/03

2. 2 Introduction DFA(difficulty factors assessment) can be used to identify what problem features (i.e., factors) facilitate problem solving. three types of knowledge for problem solving: contextual, conceptual, and procedural knowledge. Contextual → candy bar Conceptual → fraction bars Procedural → common denominator To evaluate whether each scaffold facilitated addition and subtraction of fractions, we used DFA.

3. 3 Method 223 sixth-grade students ： urban(137) 、 suburban(86) Procedure ： pretest (incorporated DFA) →implement an intervention →posttest (identical to the pretest) Pretest & Posttest ： –same denominators –unlike denominators –adding three fractions –subtracting mixed numbers –identifying a verbal description of the conventional procedure

4. 4 correct combine-both error –combine both numerator and denominator fail-to-convert error –fail to convert numerators after finding a common denominator other error

5. 5 fading of scaffolding 找公分母

6. 6 Result-pretest Average accuracy ： –All ─ 45% –Same denominator ─ 80% –unlike denominators ─ 40% –other three items ─ 37% to 42% suburban schools had higher accuracy scores than students at the urban schools at pretest (Ms = 62% vs. 35% correct), F(1, 221) = 51.9, p <.0001.

7. 7 the least the most 52% 33%

8. 8 Result-pretest Summary Each of the scaffolds reduced combine-both errors, but only the conceptual scaffold consistently reduced fail-to- convert errors.

9. 9 Result-posttest 51% 66% 43%53% 22%13%

10. 10 Result-posttest Summary children were more accurate across a range of problems made many fewer common errors such as adding the numerator and denominator had less need for the scaffolds seemed more likely to correctly use the conventional procedure.

11. 11 Discussion Why the conceptual and contextual knowledge scaffolds may have facilitated accurate problem solving? The rationale for that approach is that students need to understand the key ideas in order to have something to connect with procedural rules. Three general design suggestions emerged from integrating these findings with past research: story contexts may be useful scaffolds for introducing new tasks or problem types visual representations may facilitate problem solving scaffolding intermediate procedural steps and then fading the scaffolding may support learning and problem solving.